Alright, guys. So in the past couple of videos, we've talked about electric forces and how charges exert forces on one another through electric fields and forces things like that. In this video, we're going to cover electric potential energy. Let's check it out. So, basically, imagine I had these two identical charges and with a hold them in place, we know there's some electric force between them, so if I release them, they would both go flying opposite in opposite directions and they would gain some velocity, which means that in this process, they've gained some kinetic energy. Remember that kinetic energy is energy associated with objects in motion. But this energy couldn't have just come from nowhere. In fact, what happens is that these two charges, when they have some distance apart, they have some stored energy between them. We know that that's stored energy is called potential energy. Now, in the case of electricity, we're just going to call this electric potential energy and what we're talking about charge conservation or energy conservation. We know that anything that we lose in potential energy is gained in kinetic energy. So, for instance, if I have a charge that loses one joule of potential energy and it gains eleven joules in kinetic energy assuming that we have no non-conservative forces involved.

So I've been talking about two-point charges. So, for instance, if we had Q_{1} and Q_{2} and they were separated by some distance r, then that means that the electric potential energy between them is just going to be simply K Q_{1}Q_{2} divided by r. Another way that you might see this is if you have, like a big charge, you might see this as big Q. Little q divided by r. There are two differences I want to point out here. So one, we were talking about Coulomb's law, and this was always R squared. So we have to be very careful when we're talking about electric potential energy because this actually decreases as one over r not one over r squared like it does in Coulomb's law. You might be tempted just sort of like through habit to write one over r squared. So just be careful you're doing one over r with energy. If you've actually seen our chapter on gravitation, it's very similar to how gravitational potential energy was always one over r. But the force was one over r squared. So just in case you've seen that before, the other difference is that the signs of the charges and energy actually do matter when we're talking about electric potential. Remember that the pro tip I gave you guys for Coulomb's law was that you were just going to calculate the magnitude and then worry about the direction later. Well, here we actually are supposed to take the signs into accounts. So that's basically all we need to know about electric potential energy. Let's go ahead and take a look at a quick example.

So how far apart must a three microcoulombs and negative two microcoulombs charge be so their potential energy is something? So we have a positive and negative point charge right here, which means we're going to use our formula for electric potential energy. So we've got, let's see, we're looking for the distance r, the distance between these two charges. So now we're going to use the potential energy, and that's just going to be K Q_{1}Q_{2} divided by r. So let's just go ahead and make sure that I have everything else that I need to solve the problem. So I have the electric potential energy. This is the negative 100 millijoules. The negative actually does matter. And then I have K, which is just a constant That's the Coulomb's constant, the two charges involved, and now supposed to be finding what the distance between these charges has to be. So all you have to do is just rearrange this using some algebra. I'm going to move this r up and I have to move this u down. So basically, they're just trading places. So that means that the distance is just going to be \( \frac{8.99 \times 10^9 \times 3 \times 10^{-6} \times -2 \times 10^{-6}}{-100 \times 10^{-3}} \). And now I can go ahead and plug all the stuff in. So if you go ahead and plug this in, you should get a distance of 0.54 m so that's how far these things have to be apart from one another so that their potential energy between them is this negative 100 millijoules.

Alright, that's pretty much it. So now we've talked about two-point charges in this example. So what if we have a collection of charges? So, for instance, what if I have an arrangement of three charges? Well, what happens is that for a group of charges, we know that there exists a potential energy between any two charges. So, for instance, these two charges Q_{1} and Q_{2} have a potential energy between them. And if their distances are r_{12}, they're going to have U_{12} between them. But there's also this pair right here. There's the pair between Q_{1} and Q_{3} that's going to have a potential energy of U_{13}, and likewise, this Q_{3} and Q_{2} has the potential energy of U_{23}. So what happens is there exists this potential energy between all the pairs of charges in this assembly here. So that means that the total amount of potential energy is just going to be the sum of all those potential energy. So U_{12} plus U_{13} plus U_{23}. Now if you have more and more charges, you're just going to add more and more terms together, basically, you know, adding all of the pairs of charges together, all the potential energies between them. Most of the time, you only see about three or four anyway, because there isn't interest in being a bunch of terms to the other way. You might also see this as the energy required to separate each charge to infinity or it's the energy to bring them in from infinity. So that's two ways you might actually see that written. So let's go ahead and work out this example of how much potential energy is carried by this following system of charges. So first things first, we just want to go ahead and label all of these charges. So this is one coulomb, negative two coulombs and three coulombs. So I'm just going to call each one of these Q_{1}, Q_{2} and then Q_{3} just because that's going to be easiest. Okay, so this pair right here has a potential energy, and this is going to be U_{12} and then this pair over here is U_{23} and then finally, I have a pair between Q_{1} and Q_{3}, and all I have to do is just make sure that I'm dealing with each one of them separately. And then I need to know what this distance is as well. So the distance between Q_{1} and Q_{3}. Okay. So we know if we're trying to figure out what the total potential energy is, U total, we just have to add all these things up together. So U_{12}, then U_{13} and then U_{23}. Okay, so let's see U_{12} is just going to be \( \frac{8.99 \times 10^9 \times (1 \text{ coulomb}) \times (-2 \text{ coulombs})}{4 \text{ m}} \). Okay, so that's this term right here. Now U_{13}, I'm going to have to figure this out. So let's see. I know that this distance right here is basically the hypotenuse of the triangle between this 1 m and 3 m sides. So if I use the Pythagorean theorem, then r_{13} is just going to be 5 m. Okay, So that means that I'm going to use \( \frac{8.99 \times 10^9 \times (1 \text{ coulomb}) \times (3 \text{ coulombs})}{5 \text{ m}} \). And now, lastly, all I have to do is just add this last term right here. This is going to be \( \frac{8.99 \times 10^9 \times (-2 \text{ coulombs}) \times (3 \text{ coulombs})}{3 \text{ m}} \). You go ahead and plug all of these things in separately and then add them all up together, you should get a total potential energy of -1.\times 10^10 Joules. Alright, so that's how we work with this stuff with potential energies. It's basically just like Coulomb's law, except we just have to do one over r instead of one over r squared. And the other difference is that we don't have to do any vector addition or anything like that because these things are energies and these are scalars, they're not vectors. We don't have to do any decompositions, sine or cosine, anything like that.

Okay, let's go ahead and do a couple more examples and let's keep going, let me know if you have any questions.